ON SOME VERTEX ALGEBRAS RELATED TO ${V}_{-1}\left(\mathfrak{sl}(n)\right)$ AND THEIR CHARACTERS

Abstract

We consider several vertex operator algebras and superalgebras closely related to ${V}_{-1}\left(\mathfrak{sl}(n)\right)$, n ≥ 3 : (a) the parafermionic subalgebra K($\mathfrak{sl}$(n); −1) for which we completely describe its inner structure, (b) the vacuum algebra Ω(V₋₁($\mathfrak{sl}$(n))), and (c) an infinite extension $\mathcal{U}$ of V₋₁($\mathfrak{sl}$(n)) obtained from certain irreducible ordinary modules with integral conformal weights. It turns out that $\mathcal{U}$ is isomorphic to the coset vertex algebra $\mathfrak{psl}$(n|n)₁/$\mathfrak{sl}$(n)₁, n ≥ 3. We show that V₋₁($\mathfrak{sl}$(n)) admits precisely n ordinary irreducible modules, up to isomorphism. This leads to the conjecture that $\mathcal{U}$ is quasi-lisse.We present evidence in support of this conjecture: we prove that the (super)character of $\mathcal{U}$ is quasimodular of weight one by virtue of being the constant term of a meromorphic Jacobi form of index zero. Explicit formulas and MLDE for characters and supercharacters are given for $\mathfrak{g}$ = $\mathfrak{sl}$(3) and outlined for general n. We present a conjectural family of 2nd order MLDEs for characters of vertex algebras $\mathfrak{psl}$(n|n)₁, n ≥ 2. We finish with a theorem pertaining to characters of $\mathfrak{psl}$(n|n)₁ and $\mathcal{U}$-modules.